Dando continuidade a esta pesquisa, pretende-se melhorar o m´etodo de esti-ma¸c˜ao baseado no algoritmo EM por Monte Carlo para o modelo de regress˜ao com efeito aleat´orio substituindo o algoritmo de Metropolis-Hasting por outro m´etodo de amostragem por rejei¸c˜ao. Conseguindo melhorar esse m´etodo de estima¸c˜ao, pode-se propor medidas de influˆencia baseadas na fun¸c˜ao Q-afastamento (ZHU; LEE, 2001) para estudar medidas de influˆencia local para os modelos de regress˜ao com efeito aleat´orio.
Outra pesquisa ser´a propor um m´etodo de estima¸c˜ao para os parˆametros da distribui¸c˜ao beta Weibull modificada para dados com observa¸c˜oes censuradas, al´em de apre-sentar um modelo de regress˜ao para a distribui¸c˜ao beta Weibull modificada. Tamb´em podem ser introduzida uma an´alise de diagn´ostico (influˆencia local e res´ıduos) para esse novo modelo de regress˜ao.
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Apˆendice A - Elementos da matriz de informa¸c˜ao observada para o modelo de regress˜ao log-Weibull estendida
Lλλ = −r λ2 Lλσ = 1 σ n X i=1 zihiexp(xT i β) Lλβj = n X i=1 xijexp(xTi β) + 1 σ n X i=1 xijexp(xTi β)hi h 1 − σ exp(−zi) i Lσσ = r σ2 + 1 σ2 X i²F n zi h 2 + exp(zi)(zi+ 2) io + λ σ2 n X i=1 ziexp(xTi β)hi h − 2 − zi(1 + exp(zi)) i Lσβj = 1 σ2 X i²F xijh1 + exp(zi)(1 + zi)i+ λ σ2 n X i=1 xijexp(xTi β)hi h − 1 + zi(σ − 1 − exp(zi)) i Lβjβs = 1 σ2 X i²F xijxisexp(zi) + λ n X i=1 xijxisexp(xT i β) +λ σ2 n X i=1 xijxisexp(xT i β)hi h 2σ − 1 − σ2exp(−zi) − exp(zi) i em que j, s = 1, 2, . . . , p, hi = exp ³ zi+ exp(zi) ´ e zi = yi−xT iβ σ
Apˆendice B: Demonstra¸c˜ao do teorema 1
Para a distribui¸c˜ao log-Burr XII (15), a fun¸c˜ao geradora de momentos ´e dada pela resolu¸c˜ao da seguinte equa¸c˜ao
MY(T ) = E(exp(ty)) = Z −∞ ∞ exp(ty)kc · 1 +³exp(y) s ´c¸−k−1³exp(y) s ´c dt
Fa¸ca a seguinte mudan¸ca de vari´avel u = ³ exp(y) s ´c , ent˜ao, du = c ³ exp(y) s ´c dy. Portanto, MY(T ) = Z −∞ ∞ (su1c)tk(1 + u)−k−1du
Agora, fa¸ca outra mudan¸ca de vari´avel, considere que v = 1
1+u, ent˜ao dv = −(1 + u)−2du. E
reescreva a integral MY(T ) = Z 1 0 −stk(1 − v)ctv−t c+k−1dv kstB · t c + 1, k − t c ¸ , se kc > t
em que B(a, b) ´e a fun¸c˜ao beta completa (LAWLESS, 2003). Para obter a segunda identidade, reorganize o integrando como o n´ucleo da fun¸c˜ao de densidade da distribui¸c˜ao beta.
Apˆendice C - Elementos da matriz de informa¸c˜ao observada para o modelo de regress˜ao log-Burr XII
Lkk = − r k2 Lkσ = n X i=1 zi σhi Lkβj = n X i=1 xij σ hi Lσσ = r σ2 + 2 σ2 X i²F zi− 2(k + 1)X i²F zi σ2hi− (k + 1)X i²F ³zi σ ´2 hi+ X i²F ³zi σhi ´2 − kX i²C zi σ2hi − kX i²C ³zi σ ´2 hi+ k n X i=1 ³zi σhi ´2 Lσβj = X i²F xij σ2 − (k + 1)X i²F ³xij σ2 +xij zi σ 2´ hi+X i²F xij σ zi σhi 2 −kX i²C ³xij σ2 +xij zi σ 2´ hi+ k n X i=1 xij σ zi σhi 2 Lβjβs = −(k + 1)X i²F xijxis σ2 hi+ (k + 1)X i²F xijxis σ2 hi2− kX i²C xijxis σ2 hi+ kX i²C xijxis σ2 hi2 em que j, s = 1, 2, . . . , p, hi = exp(zi) 1+exp(zi) e zi = yi−xT iβ σ
Apˆendice D - Programa na linguagem de programa¸c˜ao matricial Ox para verificar o desempenho do modelo de regress˜ao log-Burr XII por meio do teste da raz˜ao de verossimilhan¸ca
//**PROGRAMA para simular de New Weibull e testar logistica**// #include<oxstd.h> #include<oxdraw.h> #include<oxfloat.h>
#include<maximize.h> #include<simula.h> #import<maximize> #pragma link("maximize.oxo")
static decl s_mT; //dados T simulados static decl s_mX; //covari´avel simulada static decl s_mC;//censura simulada
static decl dados; // valores de y e da censura
log_vero(const vP,const adFunc, const avScore,const amHessian) { decl n=rows(s_mT); decl cont,y,cens,xx0, xx1 ; decl uns=ones(n,1); decl vero=zeros(1,n); decl a1=fabs(vP[0][0]); //k
decl a2=fabs(vP[1][0]); //sigma decl a3=vP[2][0]; //beta 0 decl a4=vP[3][0]; //beta 1
for(cont=0;cont<n;++cont) { //y=log(dados[cont][0]);
y=log(dados[cont][0]); cens=s_mC[cont][0]; xx0=1; xx1=s_mX[cont][0]; decl xbeta=xx0*a3+xx1*a4; decl z=(y-xbeta)/a2;
if(dados[cont][1]==1)
vero[0][cont]=log(a1)-log(a2)+z-(a1+1)*log(1+exp(z)); if(dados[cont][1]==0) vero[0][cont]= -a1*log(1+exp(z)); } adFunc[0]=double(vero*uns); return 1; }
///////////////////////////////////////////////////////////////////////////// log_vero1(const vP1,const adFunc1, const avScore,const amHessian) {
decl n=rows(s_mT); decl cont,y,cens,xx0, xx1 ; decl uns=ones(n,1); decl vero1=zeros(1,n); decl a2=fabs(vP1[0][0]); //sigma
for(cont=0;cont<n;++cont) { //y=log(dados[cont][0]);
y=log(dados[cont][0]); cens=s_mC[cont][0]; xx0=1; xx1=s_mX[cont][0]; decl xbeta=xx0*a3+xx1*a4; decl z=(y-xbeta)/a2;
if(dados[cont][1]==1) vero1[0][cont]=-log(a2)+z-(1+1)*log(1+exp(z)); if(dados[cont][1]==0) vero1[0][cont]= -1*log(1+exp(z)); } adFunc1[0]=double(vero1*uns); return 1; } ///////////////////////////////////////////////////////////////////////////// main() { decl nc=30; decl dfunc; decl dfunc1; decl rep=1000; decl
inter=0; decl ncensur=zeros(rep,1); decl lambda=zeros(rep,1);
do { decl unif; unif=ranu(nc,1); s_mX=ranu(nc,1); print(" os valores da covari´avel \n", s_mX);
//gerando T
decl beta0=-3; decl beta1=3; decl uns1=ones(nc,1);
decl alpha=exp(beta0.*uns1+beta1.*s_mX[][0]); decl k=0.27; decl sigma=0.36;
s_mT=alpha.*(((1-unif).^(-1/k))-1).^sigma; decl m; decl scensur;
scensur=0; dados=ones(nc,2); s_mC=0.4*ranu(nc,1);
for (m=0; m<nc; m++) { if (s_mC[m][0]-s_mT[m][0] < 0 && scensur <3) { dados[m][0]= s_mC[m][0]; dados[m][1]=0; scensur=scensur+1; } else {dados[m][0]= s_mT[m][0]; } }
print(" os valores de T e da censura \n", dados); decl vP=<0.27;0.36;-3;3>; log_vero(vP,&dfunc,0,0); decl mhess=0.00001*unit(4);
decl ir; MaxControl(-1,20);
ir=MaxBFGS(log_vero,&vP,&dfunc,&mhess,1);
////////////////////////////////////////////////////////// decl vP1=<0.36;-3;3>; log_vero1(vP1,&dfunc1,0,0); decl ir1; MaxControl(100,20);
ir1=MaxBFGS(log_vero1,&vP1,&dfunc1,&mhess,1);
////////////////////////////////////////////////////////// if (ir == MAX_CONV && ir1 == MAX_CONV )
{ print(" os valores dos parametros \n", vP); print(" os valores dos parametros \n", vP1);
println("vero=" ,dfunc); println("vero1=" ,dfunc1); decl verologistica=dfunc1; decl veroburr=dfunc; ncensur[inter][0]=sumc(dados[][1]);
//teste da raz~ao de verossimilhan¸ca
lambda[inter][0]=-2*(verologistica-veroburr); ++inter; } } while (inter<rep);
print(" teste da razao de verossimilhanca \n", lambda); print(" n´umero de censuras na amostra \n", ncensur);
decl j; decl pvalor=ones(rep,1); for (j=0; j<rep; j++) { if (lambda[j][0] > 3.841459)
{ pvalor[j][0]= 1; } else
{pvalor[j][0]= 0; } }
decl contagem=sumc(pvalor); print("numero de amostras em que H0 ´e rejeitada \n", contagem); }
Apˆendice E - Programa na linguagem de programa¸c˜ao matricial OX para simular res´ıduo tipo martingale a partir do modelo de regress˜ao log-Weibull estendida e programa no software R para obter gr´afico de probabilidade normal dos res´ıduos simulados
//**PROGRAMA PARA CRIAR GERAR RES´IDUO TIPO MARTINGALE **// //**Modelo log-Weibull estendida**//
#include<oxstd.h> #include<oxdraw.h> #include<oxfloat.h> #include<maximize.h> #include<simula.h> #import<maximize> #pragma link("maximize.oxo")
static decl s_mT; //dados T simulados static decl s_mX; //covari´avel simulada static decl s_mC;//censura simulada
static decl dados; // valores de y e da censura
log_vero(const vP,const adFunc, const avScore,const amHessian) { decl n=rows(s_mT); decl cont,y,cens,xx0, xx1 ; decl uns=ones(n,1); decl vero=zeros(1,n); decl a1=fabs(vP[0][0]); //lambda
decl a2=fabs(vP[1][0]); //sigma decl a3=vP[2][0]; //beta 0 decl a4=vP[3][0]; //beta 1
for(cont=0;cont<n;++cont) { y=log(dados[cont][0]); cens=s_mC[cont][0]; xx0=1; xx1=s_mX[cont][0]; decl xbeta=xx0*a3+xx1*a4; decl z=(y-xbeta)/a2;
if(dados[cont][1]==1) vero[0][cont]=log(a1+0.00000000001)
-log(a2+0.00000000001)+xbeta+z+exp(z)+a1*exp(xbeta)*(1-exp(exp(z))); if(dados[cont][1]==0) vero[0][cont]= a1*exp(xbeta)*(1-exp(exp(z))); }
adFunc[0]=double(vero*uns); return 1; }
main() { decl nc=30; decl dfunc; decl rep=1000; decl res=zeros(nc,rep); decl residuos=zeros(nc,rep); decl
ordelinha=zeros(nc,rep); decl inter=0; decl ncensur=zeros(rep,1); do {
decl unif; unif=ranu(nc,1); print(" os valores DA UNIFORME \n", unif); s_mX=ranu(nc,1); print(" os valores da covari´avel \n", s_mX); //gerando T
decl beta0=-3; decl beta1=3; decl uns1=ones(nc,1); decl
alpha=exp(beta0.*uns1+beta1.*s_mX[][0]); decl lambda=0.1; decl sigma=2;
s_mT=( ((alpha).^(1/sigma)).*log(1-(log(1-unif)./(lambda.*alpha))) ).^(sigma);
print(" os valores simulados \n", s_mT);
decl m; decl scensur; scensur=0; dados=ones(nc,2);
s_mC=0.5*ranu(nc,1); print(" os valores simulados da censura \n", s_mC); for (m=0; m<nc; m++) { if (s_mC[m][0]-s_mT[m][0] < 0 && scensur < 3) { dados[m][0]= s_mC[m][0]; dados[m][1]=0; scensur=scensur+1; } else {dados[m][0]= s_mT[m][0]; } }
print("n´umero de censurados \n", scensur); print(" os valores de T e da censura \n", dados);
decl vP=<0.1;2;-3;3>; log_vero(vP,&dfunc,0,0); println("vero=" ,dfunc);
decl mhess=0.00001*unit(4); decl ir; MaxControl(-1,20);
ir=MaxBFGS(log_vero,&vP,&dfunc,&mhess,1); if (ir == MAX_CONV) { print(" os valores dos parametros \n", vP);
ncensur[inter][0]=sumc(dados[][1]); decl xbeta1=(vP[2][0]*uns1+
decl zz=(log(s_mT[][0])-xbeta1); decl zzz=zz./fabs(vP[1][0]); decl sob2=1-exp(exp(zzz)); decl
sob3=exp(fabs(vP[0][0]).*exp(xbeta1).*sob2); print(" os valores de sob3 \n", sob3);
decl martingal= dados[][1]+log(sob3+0.000000000001); res[][inter]= (martingal./fabs(martingal)).*
(1.4142*(-martingal-(dados[][1].*log(dados[][1]-martingal))).^(1/2));; decl resorde=(sortc(res[][inter])); residuos[][inter]=resorde;
++inter; } } while (inter<rep);
print(" os valores dos residuos simulados \n", res); print(" n´umero de censuras na amostra \n", ncensur);
decl j; decl mresults = zeros (nc,3); for (j=0; j < nc; ++j) { // calculando estat´ısticas
mresults[j][0]=min(residuos[j][]); mresults[j][1]=meanr(residuos[j][]);
mresults[j][2]=max(residuos[j][]); } print(" estat´ısticas dos valores simulados \n", mresults); print(" n´umero de itera¸c~oes \n", inter); }
library(RODBC)
x1<-seq(-3,3,0.01) y1<-x1
# Read EXCEL sheet containing spatial data channel
<-odbcConnectExcel("MARTI_50_SC.xls") MARTISC <- sqlFetch(channel, "MARTI_50_SC") odbcClose(channel)
channel odbcConnectExcel("DEVI_50_SC.xls") DEVISC <-sqlFetch(channel, "DEVI_50_SC") odbcClose(channel) q1<-qqnorm(MARTISC$res50ESC,plot.it = FALSE)
q2<-qqnorm(DEVISC$res50ESC,plot.it = FALSE)
plot(range(q1$x, q2$x), range(q1$y, q2$y), type = "n",,main = "", xlab="Percentis da N(0,1)",ylab="Res´ıduo") points(q1,col =
"blue",pch=1) points(q2, col = "red", pch = 3) lines(x1,y1) texto<-expression(r[M],r[D])
legend("bottomright",texto,text.col=c("blue","red"),col=c("blue","red"),pch = c(1,3), cex = 1,bty = "o" )
mtext("0% de censura", cex = 0.8, line =1,side=3)
legend("topleft",expression(k=="0,27"), cex = 1,bty = "n")
Apˆendice F - Programa na linguagem de programa¸c˜ao matricial OX para estimar os parˆametros do modelo de regress˜ao log-Weibull estendida ajustado aos dados dos peixes da esp´ecie Golden Shiner usando o m´etodo de m´axima verossimilhan¸ca e calcular os res´ıduos
//**PROGRAMA PARA ESTIMAR OS PAR^AMETROS DO MODELO LOG NOVA WEIBULL ESTENDIDA**//
//**DADOS REAIS - PEIXES DA EXP´ECIE GOLDEN SHINER**// #include<oxstd.h> #include<oxdraw.h> #include<oxfloat.h>
#include<maximize.h> #include<simula.h> #pragma link("maximize.oxo") static decl g_mY; static decl g_mX;
log_vero(const vP,const adFunc, const avScore,const amHessian) { decl n=rows(g_mY); decl cont,y,xx0, xx1, xx2, xx3, xx4,xx5, xx6, xx7; decl uns=ones(n,1); decl vero=zeros(1,n); decl
a1=fabs(vP[0][0]); //lambda decl a2=fabs(vP[1][0]); //sigma decl a3=vP[2][0]; //beta 0 decl a4=vP[3][0]; //beta 1 decl a5=vP[4][0]; //beta 2 decl a6=vP[5][0]; //beta 3 decl a7=vP[6][0]; //beta 4 decl a8=vP[7][0]; //beta 5 decl a9=vP[8][0]; //beta 6 decl a10=vP[9][0]; //beta 7 for(cont=0;cont<n;++cont) { y=g_mY[cont][0]; xx0=g_mY[cont][2]; xx1=g_mY[cont][3]; xx2=g_mY[cont][4]; xx3=g_mY[cont][5];
xx4=g_mY[cont][6]; xx5=g_mY[cont][7]; xx6=g_mY[cont][8]; xx7=g_mY[cont][9];
decl z=(y-(xbeta))/a2;
if(g_mY[cont][1]==1) vero[0][cont]=log(a1+0.00000000001)
-log(a2+0.00000000001)+xbeta+z+exp(z)+a1*exp(xbeta)*(1-exp(exp(z))); if(g_mY[cont][1]==0) vero[0][cont]= a1*exp(xbeta)*(1-exp(exp(z))); } adFunc[0]=double(vero*uns);
return 1; } main() {
g_mY = loadmat("c:\TESE\New_Weibull\especie22.txt") ; //print(" Dados",g_mY);
decl nc=rows(g_mY); decl dfunc; decl avScore; decl
vP=<0.5;3;5.3439;1.9119;0.1054;-0.1229;0.0367;0.0364;0.2627;-0.09434>; log_vero(vP,&dfunc,&avScore,0); println("vero=" ,dfunc);
MaxControl(-1,20); decl mhess=0.00001*unit(4); decl ir, var; ir=MaxBFGS(log_vero,&vP,&dfunc,&mhess,1);
Num1Derivative(log_vero,vP,&avScore);
Num2Derivative(log_vero,vP,&mhess); var=invertsym((-1)*mhess); print(" os valores das vari^ancias \n",var); print(" os valores dos parametros \n", vP); print(" os valores score \n", avScore);
decl zz=(g_mY[][0]-(vP[2][0].*g_mY[][2]+ vP[3][0].*g_mY[][3]+ vP[4][0].*g_mY[][4]+ vP[5][0].*g_mY[][5]+ vP[6][0].*g_mY[][6]+ vP[7][0].*g_mY[][7]+ vP[8][0].*g_mY[][8]+ vP[9][0].*g_mY[][9])); print(" os valores de zz \n", zz); decl xbeta1=(vP[2][0].*g_mY[][2]+ vP[3][0].*g_mY[][3]+ vP[4][0].*g_mY[][4]+ vP[5][0].*g_mY[][5]+ vP[6][0].*g_mY[][6]+ vP[7][0].*g_mY[][7]+ vP[8][0].*g_mY[][8]+ vP[9][0].*g_mY[][9]); print(" os valores de xbeta1 \n", xbeta1);
decl res=zz./vP[1][0]; print(" os valores dos residuos \n", res); decl sob1=exp(res); print(" os valores de exp(res) \n", sob1); decl sob2=1-exp(sob1);
print(" os valores de 1-exp(exp(res)) \n", sob2); decl sob3=exp(vP[0][0].*exp(xbeta1).*sob2);
print(" os valores de sob estimada para y\n", sob3);
decl martingale= g_mY[][1]+log(sob3); print(" os valores do residuo martingale \n", martingale);
decl tipmartin= (martingale./fabs(martingale)).*
(1.4142*(-martingale-(g_mY[][1].*log(g_mY[][1]-martingale))).^(1/2)); print(" os valores do residuo tipo martingale \n", tipmartin);
}
Apˆendice G - Programa na linguagem de programa¸c˜ao matricial OX para obter as estimativas de jackknife e medidas de influˆencia global para o modelo de regress˜ao log-Weibull estendida ajustado aos dados dos peixes da esp´ecie Golden Shiner //**PROGRAMA PARA ESTIMAR OS PAR^AMETROS DO MODELO DE REGRESS~AO**\\
//**LOG-NOVA WEIBULL ESTENDIDA - M´etodo de Jackknife**// //**MEDIDAS DE INFLU^ENCIA GLOBAL **//
//**DADOS PEIXES DA ESP´ECIE GOLDEN SHINER**// #include<oxstd.h> #include<oxdraw.h> #include<oxfloat.h>
#include<maximize.h> #include<simula.h> #pragma link("maximize.oxo") static decl g_mY; static decl g_mX;
log_vero(const vP,const adFunc, const avScore,const amHessian) { decl n=rows(g_mY);
decl cont,t,xx0, xx1, xx2, xx3, xx4,xx5, xx6, xx7; decl uns=ones(n,1); decl vero=zeros(1,n);
decl a3=vP[2][0]; //beta 0 decl a4=vP[3][0]; //beta 1 decl a5=vP[4][0]; //beta 2 decl a6=vP[5][0]; //beta 3 decl a7=vP[6][0]; //beta 4 decl a8=vP[7][0]; //beta 5 decl a9=vP[8][0]; //beta 6 decl a10=vP[9][0]; //beta 7 for(cont=0;cont<n;++cont) {
//t=log(g_mY[cont][0]);
t=g_mY[cont][0]; xx0=g_mY[cont][2]; xx1=g_mY[cont][3]; xx2=g_mY[cont][4]; xx3=g_mY[cont][5]; xx4=g_mY[cont][6]; xx5=g_mY[cont][7]; xx6=g_mY[cont][8]; xx7=g_mY[cont][9];
decl xbeta=xx0*a3+xx1*a4+xx2*a5+xx3*a6+xx4*a7+xx5*a8+xx6*a9+xx7*a10;